Final answer:
The Moon's orbital period of about 0.08 years and its average distance from Earth of 0.0027 AU can be used with Kepler's third law to estimate Earth's mass relative to the Sun. Consideration of the gravitational constant and the negligible mass of the Moon in comparison to Earth allows for the simplification of the formula.
Step-by-step explanation:
Calculating Earth's Mass Relative to the Sun Using Kepler's Third Law
The Moon takes about 0.08 years (or approximately 1 month) to complete an orbit around Earth, which is a distance of approximately 400,000 km (0.0027 AU). By applying Kepler's third law, as modified by Newton, we can estimate the mass of Earth in relation to the Sun's mass.
Kepler's third law states that the square of the orbital period (T) is directly proportional to the cube of the semi-major axis of the orbit (a). In the case of the Moon orbiting the Earth, this period T is 0.08 years, and the average distance a is 0.0027 AU.
The modified version of Kepler's law includes the gravitational constant (G) and is expressed as: T^2 = (4π^2/G(M₁+M₂)) * a^3, where M₁ is the mass of Earth and M₂ is the mass of the Moon. However, because M₂ is much smaller than M₁, it can be neglected, simplifying the equation to calculate Earth's mass relative to the Sun.
Given that the mass of the Moon is significantly smaller than Earth's mass, this calculation will give us a good estimate of Earth's mass relative to the Sun's mass. Note that careful consideration of the units used in these calculations is crucial to obtaining a correct result.